| L(s) = 1 | − 2.42·2-s + 3-s + 3.86·4-s + 1.50·5-s − 2.42·6-s + 7-s − 4.52·8-s + 9-s − 3.64·10-s − 2.29·11-s + 3.86·12-s − 3.95·13-s − 2.42·14-s + 1.50·15-s + 3.21·16-s + 6.76·17-s − 2.42·18-s + 6.60·19-s + 5.82·20-s + 21-s + 5.56·22-s + 8.27·23-s − 4.52·24-s − 2.73·25-s + 9.58·26-s + 27-s + 3.86·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.673·5-s − 0.988·6-s + 0.377·7-s − 1.59·8-s + 0.333·9-s − 1.15·10-s − 0.692·11-s + 1.11·12-s − 1.09·13-s − 0.647·14-s + 0.388·15-s + 0.803·16-s + 1.64·17-s − 0.570·18-s + 1.51·19-s + 1.30·20-s + 0.218·21-s + 1.18·22-s + 1.72·23-s − 0.922·24-s − 0.546·25-s + 1.88·26-s + 0.192·27-s + 0.730·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.296014433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.296014433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 - 6.76T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 8.27T + 23T^{2} \) |
| 29 | \( 1 + 0.934T + 29T^{2} \) |
| 31 | \( 1 - 4.90T + 31T^{2} \) |
| 37 | \( 1 - 0.901T + 37T^{2} \) |
| 41 | \( 1 + 7.22T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 - 0.887T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 - 2.31T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 + 9.78T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.363948516231940722961960543102, −7.921473435393081638307075179591, −7.29654468134070316843920201954, −6.73453972485296240690482766174, −5.42916611911898892970974027751, −5.01725899057939553611193358249, −3.31015852553314422556759833947, −2.65212376002812481979033274848, −1.72050651104226210571835710771, −0.857980196045327642415352177928,
0.857980196045327642415352177928, 1.72050651104226210571835710771, 2.65212376002812481979033274848, 3.31015852553314422556759833947, 5.01725899057939553611193358249, 5.42916611911898892970974027751, 6.73453972485296240690482766174, 7.29654468134070316843920201954, 7.921473435393081638307075179591, 8.363948516231940722961960543102
